Circumference and Perimeter of a Sector
Theory
The circumference of a circle is \(C = 2\pi r = \pi d\). This Year 9 maths page revises the Year 8 circle work and extends it to arcs and the perimeter of a sector, shapes with curved edges, exact answers, and working back from a circumference to the radius.
The radius \(r\) runs from the centre of a circle to its edge. The diameter \(d\) goes right across through the centre, so \(d = 2r\).
The circumference is the distance all the way round the circle. It is the circle's perimeter.
\(\pi\) (pi) is the number of times the diameter fits around the circumference, about \(3.14159\). Use the \(\pi\) key on your calculator, unless a question tells you to use \(3.14\) or \(\dfrac{22}{7}\).
A sector is a slice of a circle cut by two radii, with an angle \(\theta\) at the centre. Its curved edge is an arc. A semicircle is a sector of \(180^\circ\) and a quadrant is a sector of \(90^\circ\).
An exact answer keeps \(\pi\) as a symbol, such as \(22\pi\) cm. A rounded answer uses the \(\pi\) key and rounds once, at the end.
A sector with angle \(\theta\) is \(\dfrac{\theta}{360}\) of the circle:
| Shape | Curved edge | Perimeter |
|---|---|---|
| circle | \(2\pi r\) | \(2\pi r\) |
| semicircle | \(\pi r\) | \(\pi r + 2r\) |
| quadrant | \(\dfrac{1}{2}\pi r\) | \(\dfrac{1}{2}\pi r + 2r\) |
| sector of \(\theta\) | \(\dfrac{\theta}{360} \times 2\pi r\) | arc \(+ \, 2r\) |
How to find a circumference or a sector perimeter
- Find the radius. If you are given the diameter, halve it (or use \(C = \pi d\)).
- Decide the fraction of a circle: whole, half, quarter, or \(\dfrac{\theta}{360}\).
- Find the curved edge: that fraction of \(2\pi r\).
- Add the straight edges for a perimeter: two radii for a sector, the diameter for a semicircle, or the sides of the shape.
- Give the answer exactly (leave \(\pi\) in) or round once at the end, as the question asks.
(i) exactly, in terms of \(\pi\)
(ii) correct to two decimal places.
(i) With the diameter, use \(C = \pi d\). Leave \(\pi\) as a symbol.
| \(C\) | \(=\) | \(\pi \times 22\) |
| \(\) | \(=\) | \(\mathbf{22\pi} \text{ cm}\) |
(ii) Now use the \(\pi\) key and round at the end.
| \(C\) | \(=\) | \(69.115\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{69.12} \text{ cm}\) |
Using \(C = 2\pi r\) with \(r = 11\) gives the same answer.
The radius is \(7\) cm. The perimeter is the curved edge, half a circumference, plus the straight diameter.
| \(P\) | \(=\) | \(\dfrac{1}{2} \times 2\pi \times 7 + 14\) |
| \(\) | \(=\) | \(7\pi + 14\) |
| \(\) | \(=\) | \(21.991\ldots + 14\) |
| \(\) | \(\approx\) | \(\mathbf{35.99} \text{ cm}\) |
Leaving out the diameter gives only the curved edge, \(21.99\) cm, not the perimeter.
(i) its arc length
(ii) its perimeter, each correct to two decimal places.
The sector is \(\dfrac{45}{360} = \dfrac{1}{8}\) of the whole circle.
(i) Take that fraction of the circumference.
| \(\text{arc}\) | \(=\) | \(\dfrac{1}{8} \times 2\pi \times 10\) |
| \(\) | \(=\) | \(7.853\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{7.85} \text{ cm}\) |
(ii) Add the two straight radii.
| \(P\) | \(=\) | \(7.853\ldots + 10 + 10\) |
| \(\) | \(\approx\) | \(\mathbf{27.85} \text{ cm}\) |
Keep the full arc value in the calculator until the end, then round once.
(ii) A circular garden bed has a circumference of \(50\) m. Find its radius, correct to two decimal places.
(i) The two semicircles together make one whole circle with a diameter of \(60\) m. The dashed lines are inside the track, so they are not counted.
| \(P\) | \(=\) | \(2 \times 100 + \pi \times 60\) |
| \(\) | \(=\) | \(200 + 188.495\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{388.50} \text{ m}\) |
(ii) Substitute into \(C = 2\pi r\), then divide both sides by \(2\pi\).
| \(2\pi r\) | \(=\) | \(50\) |
| \(r\) | \(=\) | \(\dfrac{50}{2\pi} = 7.957\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{7.96} \text{ m}\) |
Common pitfalls
Frequently asked questions
What is the formula for the circumference of a circle?
\(C = 2\pi r\), or \(C = \pi d\). A circle with a diameter of \(22\) cm has a circumference of \(22\pi \approx 69.12\) cm.
How do you find the perimeter of a semicircle?
Add half the circumference to the diameter: \(P = \pi r + 2r\). For a diameter of \(14\) cm, \(P = 7\pi + 14 \approx 35.99\) cm.
How do you find the arc length of a sector?
Multiply the full circumference, \(2\pi r\), by \(\dfrac{\theta}{360}\), where \(\theta\) is the angle at the centre in degrees.
How do you find the perimeter of a sector?
Find the arc length, then add the two radii: \(P = \dfrac{\theta}{360} \times 2\pi r + 2r\).
How do you find the radius from the circumference?
Divide the circumference by \(2\pi\): \(r = \dfrac{C}{2\pi}\). A circumference of \(50\) m gives \(r \approx 7.96\) m.
What does an exact answer in terms of pi mean?
Leave \(\pi\) as a symbol instead of using a decimal, for example \(22\pi\) cm. It is the most precise form of the answer.